## Functions of Many Variables

Functions with more than one independent variable.

y=f(x) => one independent variable.

Not just price, but incomes, ect. are independent.

z=f(x, y) Here, z = dependent, x & y = independent.

Example:

z=100-2x+5y

z=3x** ^{2}**-9y

z=e** ^{2x}**+3y Expediential fn.

Example 1: z=150-2x-3y

X-Intercept (z=0, y=0)

0= 150-2x-3(0)

=> x=150/2 =75 (75, 0, 0)

Y-Intercept (x=0, z=0)

0= 150-2(0)-3y

y= 150/3 = 50

Z-Intercept (x=0, y=0)

z= 150

How dependent variable will a change as a result of the change in independent variables.

Use partial derivatives z=f(x, y)

Terms: Change: Δ, Delta: δ

To what effect a change in x has on z while holding y constant.

To what effect a change in y has on z while holding x constant.

Example 1:

z=x** ^{3}**+3x

**y**

^{2}**+y**

^{2}

^{3}z=x** ^{3}**+[3y

**]x**

^{2}**+[y**

^{2}**]**

^{3}fx= 3x** ^{2}**+(3y

**)(2)x+0**

^{2}=3x** ^{2}**+6xy

**Constant Multiplication Remains.**

^{2}fy= 0+(3x** ^{2}**)(2)y+3y

**->**

^{2}

^{2 = (}

^{3-}

^{2) }=6x** ^{2}**y+3y

^{2}fx Slope of fn with respect to x.

fy Slope of fn with respect to y.

Example 2:

z= x** ^{0.5}**y

**-10**

^{0.5}fx= 0.5x-** ^{0.5}**(x

**)(0.5y)-10**

^{0.5}fx=(y** ^{0.5}**)(0.5)x-

**-0**

^{0.5}=0.5x-** ^{0.5}** y

^{0.5}y=nx^{n-1}

When x=25, y=9

fx=0.5(25)-** ^{0.5}**(9)

**= 3/10**

^{0.5}fy= x** ^{0.5}**(0.5)y-

^{0.5}=(0.5)x** ^{0.5}**y-

**_____Partial Derivative**

^{0.5}Plug in values= (0.5)(25)** ^{0.5}**(9)-

**= 5/6**

^{0.5}Example 3: z=3x** ^{2}**y

^{3}fx= (3y** ^{3}**)(2)x = 6xy

^{3}fy= (3x** ^{2}**)3y

**= 9x**

^{2}**y**

^{2}

^{2}Example 4: z=5x** ^{3}**-3x

**y**

^{2}**+7y**

^{2}

^{5}fx= (15x** ^{2}**)-6xy

^{2}fy= -6x** ^{2}**y+35y

^{4}

Example 5: z=(3x+5)(2x+6y)

fx= (3)(2x+6y)+(2)(3x+5)

Take derivative of 1st term with respect to x. Do not differentiate 2nd term of multiplication.

=6+18y+6x+10

=12x+18y+10

fy=(0)(2x+6y)+(6)(3x+5)

=18x+30

Example 6: Q=36KL-2K** ^{2}**-3L

**Production Function**

^{2}

fL= 36K-0-6L

=36K-6L

MPK= 36K-6K

Featured image supplied from Unsplash.

*Copyright ©* 2016 Zoë-Marie Beesley

Licensed under a Creative Commons Attribution 4.0 International License.